Orthogonal frequency division multiplexing (OFDM) is a multi-carrier modulation technique that has been widely used in digital communication systems. It has many advantages such as high spectral efficiency, resiliency to radio frequency (RF) interference, lower multi-path distortion, and high bit rates. However, OFDM suffers a high peak to average power ratio (PAPR) at transmitted signals, which causes significant signal degradation and thus reduces transmission efficiency. It is desirable to develop methods and apparatuses for reducing PAPRs in OFDM systems.